Problem 18 - Algebra 2 and Trigonometry

Question

a. Find the exact value of \(\cos 75^{\circ}\) by using \(\cos \left(45^{\circ}+30^{\circ}\right)\) b. Use the value of \(\cos 75^{\circ}\) found in a to find \(\cos 255^{\circ}\) by using \(\cos \left(180^{\circ}+75^{\circ}\right)\) c. Use \(\cos A=\sin \left(90^{\circ}-A\right)\) to find the exact value of \(\sin 15^{\circ} .\)

Step-by-Step Solution

Answer

a. \(\cos 75^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}\). b. \(\cos 255^{\circ} = \frac{-\sqrt{6} + \sqrt{2}}{4}\). c. \(\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Step 1: Formula for Angle Addition

To find the exact value of \(\cos 75^{\circ}\), we use the formula for the cosine of the sum of two angles: \(\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\). Here, \(\alpha = 45^{\circ}\) and \(\beta = 30^{\circ}\).

Step 2: Calculate Individual Trigonometric Values

Calculate the trigonometric values:- \(\cos 45^{\circ} = \frac{\sqrt{2}}{2}\)- \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\)- \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\)- \(\sin 30^{\circ} = \frac{1}{2}\)

Step 3: Substitute and Simplify

Substitute these values into the formula:\[\cos 75^{\circ} = \left( \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} \right) - \left( \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \right)\]\[= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}\]

Step 4: Calculate \(\cos 255^{\circ}\) Using Cosine Sum Formula

To find \(\cos 255^{\circ}\), use the identity \(\cos(180^{\circ} + A) = -\cos A \). Therefore:\[\cos 255^{\circ} = -\cos 75^{\circ} = -\left(\frac{\sqrt{6} - \sqrt{2}}{4}\right)\]\[= \frac{-\sqrt{6} + \sqrt{2}}{4}\]

Step 5: Use Co-function Identity to Find \(\sin 15^{\circ}\)

Use the identity \(\cos A = \sin (90^{\circ} - A)\). Here \(A = 75^{\circ}\), so \(\sin(15^{\circ}) = \cos(75^{\circ})\).Thus, \(\sin 15^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4}\).

Key Concepts

Angle Addition Formula

The Angle Addition Formula is a vital tool in trigonometry used to find the sine or cosine of the sum or difference of two angles. For cosine, this formula is given by:

• \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

This equation allows us to break down complex angles into simpler, known angles. For instance, to calculate \( \cos 75^{\circ} \), we can rewrite 75 degrees as the sum of 45 degrees and 30 degrees. These angles are well-known in trigonometry, making computation straightforward.

Using the addition formula, you simply plug in the values for the trigonometric functions of 45 and 30 degrees into the formula. This approach dramatically simplifies solving problems involving less common angles.

Cosine

Cosine is one of the primary trigonometric functions, often represented as \( \cos \). It relates the adjacent side to the hypotenuse in a right triangle. The wave-like nature of the cosine function allows for periodic repetitions, which helps in solving problems involving angles beyond the typical 0 to 90 degrees.

For common angles, we have standard values:

• \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)

These precise values are used in computations involving the Angle Addition Formula. For the exercise in question, when dealing with \( \cos 255^{\circ} \), we use the identity \( \cos(180^{\circ} + A) = -\cos A \). This helps us manage angles outside the basic quadrant range by converting them into familiar angles we can easily compute.

Sine

The sine function, represented by \( \sin \), is another cornerstone of trigonometry. It relates the opposite side to the hypotenuse in a right triangle and, like cosine, has a periodic wave form. The sine values are crucial in calculations involving Angle Addition Formulas:

• \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
• \( \sin 30^{\circ} = \frac{1}{2} \)

These values work alongside cosine values to provide solutions for trigonometric identities. In our exercise, the sine values are necessary to accurately complete the Angle Addition Formula, simplifying calculations involving angles like 75 degrees.

Co-function Identity

The Co-function Identity is a trigonometric rule that shows the close relationship between sine and cosine. It provides a convenient way to express sine in terms of cosine and vice versa for complementary angles:

• \( \cos A = \sin (90^{\circ} - A) \)

This identity reveals how one function mirrors the other across an interchanging angle difference from 90 degrees. For solving the exercise part involving \( \sin 15^{\circ} \), you can apply this identity to link it directly to \( \cos 75^{\circ} \). Such relationships allow for simpler computations of trigonometric values where direct calculation might be cumbersome.

Using these co-functions identities strategically bridges gaps, enhancing problem-solving skills in trigonometry.